Correction to: Free products of n-tuple semigroups

Ukrainian Mathematical Journal - 1) The affiliation of the first author should read Taras Shevchenko Luhansk National University, Starobilsk, Ukraine.     

Automorphisms of the endomorphism semigroup of a free commutative dimonoid

We determine all isomorphisms between the endomorphism semigroups of free commutative dimonoids and prove that all automorphisms of the endomorphism semigroup of a free commutative dimonoid are quasi-inner. In particular, we answer a question of B. I. Plotkin.     

Structure of relatively free dimonoids

Loday introduced the notion of a dimonoid and constructed the free dimonoid. The paper concerns the variety theory of dimonoids. We recall and summarize the results obtained by Loday, the author as well as others on the structure of some relatively free dimonoids. This part of the paper should be viewed as a survey, however we include some new ideas in the last sections. Namely, we construct a free (?z;rs)-dimonoid, a free (rs;rz)-dimonoid, a free rs-dimonoid, a free (rb;rs)-dimonoid, a free (rs;rb)-dimonoid and characterize the least (?z;rs)-congruence, the least (rs;rz)-congruence, the least rs-congruence, the least (rb;rs)-congruence and the least (rs;rb)-congruence on the free dimonoid. We also establish that the automorphism groups of constructed free algebras are isomorphic to the symmetric group and the semigroups of the free rs-dimonoid are anti-isomorphic.     

Some congruences on trioids

We present the least idempotent congruence on the trioid with a commutative operation, the least semilattice congruence on the trioid with an idempotent operation, and the least separative congruence on the trioid with a commutative operation. Also we construct different examples of trioids.     

Endotypes of partial equivalence relations

We classify all partial equivalence relations according to their endotype with respect to endotopisms and compute the cardinalities of endotopism semigroups (strong endotopism monoids, autotopism groups) of partial equivalence relations on a finite set.

     

Representations of the Strong Endomorphism Monoid of Finite n-Uniform Hypergraphs

In this paper, two exact representations of the strong endomorphism monoid of an arbitrary finite n-uniform hypergraph are described. It is proved that the strong endomorphism monoid of a finite n-uniform hypergraph is regular. We find all nonnegative integers n such that n-uniform hypergraphs are determined by their strong endomorphisms.     

Structure of free strong doppelsemigroups

We consider strong doppelsemigroups which are sets with two binary associative operations satisfying axioms of strong interassociativity. Commutative dimonoids in the sense of Loday are examples of strong doppelsemigroups and two strongly interassociative semigroups give rise to a strong doppelsemigroup. The main aim of this paper is to construct a free strong doppelsemigroup, a free n-dinilpotent strong doppelsemigroup, a free commutative strong doppelsemigroup and a free n-nilpotent strong doppelsemigroup. We also characterize the least n-dinilpotent congruence, the least commutative congruence, the least n-nilpotent congruence on a free strong doppelsemigroup and establish that the automorphism group of every constructed free algebra is isomorphic to the symmetric group.     

The Endotopism Semigroups of a Partial Equivalence Relation

Siberian Mathematical Journal - We study the endotopism semigroups of a nonreflexive partial equivalence. We describe the existence conditions for six types of endotopisms and...